论文标题
关于Kato的平滑效果，用于Zakharov-Kuznetsov方程的分数版本
On Kato's smoothing effect for a fractional version of the Zakharov-Kuznetsov equation
论文作者
论文摘要
在这项工作中，我们研究了一些与初始值问题（IVP）\ BEGIN {equation} \ label {main1} \ left \ left \ {\ begin {array} {ll} {ll} \ partial_ { U+U \ partial_ {x_ {1}} u = 0，\ Quad 0 <α\ leq 2，＆\\ u（x，0）= u_ {0}（x）（x），\ quad x =（x_ {1}，x_ {1}，x__ {2}，\ dots，\ dots，\ dots，\ dots，\ dots，\ dots，x _ {n} n} n} n \ geq 2，\ quad t \ in \ mathbb {r}，＆\\ \ end {array} \ right。 \ end {equation}其中$（ - δ）^{α/2} $表示$ n- $ dimensional分数laplacian。 我们表明，在合适的Sobolev空间中使用初始数据的IVP（0.1）的解决方案在$ \fracα{2} $衍生物的空间变量中表现出局部平滑效果，几乎在及时到达任何地方。试图获得这种正规化效应的主要困难之一是考虑到的操作员是非本地的，我们试图描述的属性是本地的，因此需要新的想法。尽管如此，为避免这些问题，我们使用一个扰动参数代替$（δ）^{\fracα{2}} $，通过$（i-δ）^{\fracα{2}} $，$，通过使用伪分别的callus允许所有$ freac $ freac n of pseudo-divertial Callus，我们可以通过$ \ frac of freac of freac freac freac freac freac niv freac niv freac。方向。 作为副产品，我们使用这种特殊的平滑效果来表明，欧几里得空间某些杰出子集的初始数据的额外规律性通过无限速度的流溶液传播。
In this work we study some regularity properties associated to the initial value problem (IVP) \begin{equation}\label{main1} \left\{ \begin{array}{ll} \partial_{t}u-\partial_{x_{1}}(-Δ)^{α/2} u+u\partial_{x_{1}}u=0, \quad 0< α\leq 2,& \\ u(x,0)=u_{0}(x),\quad x=(x_{1},x_{2},\dots,x_{n})\in \mathbb{R}^{n},\, n\geq 2,\quad t\in\mathbb{R},& \\ \end{array} \right. \end{equation} where $(-Δ)^{α/2}$ denotes the $n-$dimensional fractional Laplacian. We show that solutions to the IVP (0.1) with initial data in a suitable Sobolev space exhibit a local smoothing effect in the spatial variable of $\fracα{2}$ derivatives, almost everywhere in time. One of the main difficulties that emerge when trying to obtain this regularizing effect underlies that the operator in consideration is non-local, and the property we are trying to describe is local, so new ideas are required. Nevertheless, to avoid these problems, we use a perturbation argument replacing $(-Δ)^{\fracα{2}}$ by $(I-Δ)^{\fracα{2}},$ that through the use of pseudo-differential calculus allows us to show that solutions become locally smoother by $\fracα{2}$ of a derivative in all spatial directions. As a by-product, we use this particular smoothing effect to show that the extra regularity of the initial data on some distinguished subsets of the Euclidean space is propagated by the flow solution with infinity speed.
